3
$\begingroup$

Is it already known that there are infinitely many primes of the form $2n^2 + 2n + 1$? I was searching online for any articles about it but I can't find any so I suppose this is still unknown. I found that there are infinitely many such primes and one result is that there are infinitely many primes of the form $4t_n + 1$ where $t_n$ are triangular numbers.

$\endgroup$
4
$\begingroup$

Definitely out-of-reach. $$p=2x^2+2x+1\quad\Longleftrightarrow\quad 2p-1 = (2x-1)^2 $$ hence you claim is equivalent to

There are infinite values of $n\in\mathbb{N}$ such that $n^2+1$ is twice a prime

but

There are infinite values of $n\in\mathbb{N}$ such that $n^2+1$ is a prime

is still a conjecture, namely Landau's conjecture. The closest theorem we have at the moment is due to Iwaniec and Friedlander: there are an infinite number of primes of the form $a^2+b^4$.

$\endgroup$
  • $\begingroup$ there was indeed an error in my proof. I thought and assumed every primitive pythagorean triple is unique*** $\endgroup$ – unknownMe Nov 17 '16 at 8:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.